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  <div class="section" id="numpy-fft-hfft">
<h1>numpy.fft.hfft<a class="headerlink" href="#numpy-fft-hfft" title="Permalink to this headline">¶</a></h1>
<dl class="function">
<dt id="numpy.fft.hfft">
<code class="sig-prename descclassname">numpy.fft.</code><code class="sig-name descname">hfft</code><span class="sig-paren">(</span><em class="sig-param">a</em>, <em class="sig-param">n=None</em>, <em class="sig-param">axis=-1</em>, <em class="sig-param">norm=None</em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/numpy/numpy/blob/v1.18.1/numpy/fft/_pocketfft.py#L478-L566"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#numpy.fft.hfft" title="Permalink to this definition">¶</a></dt>
<dd><p>Compute the FFT of a signal that has Hermitian symmetry, i.e., a real
spectrum.</p>
<dl class="field-list">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl>
<dt><strong>a</strong><span class="classifier">array_like</span></dt><dd><p>The input array.</p>
</dd>
<dt><strong>n</strong><span class="classifier">int, optional</span></dt><dd><p>Length of the transformed axis of the output. For <em class="xref py py-obj">n</em> output
points, <code class="docutils literal notranslate"><span class="pre">n//2</span> <span class="pre">+</span> <span class="pre">1</span></code> input points are necessary.  If the input is
longer than this, it is cropped.  If it is shorter than this, it is
padded with zeros.  If <em class="xref py py-obj">n</em> is not given, it is taken to be <code class="docutils literal notranslate"><span class="pre">2*(m-1)</span></code>
where <code class="docutils literal notranslate"><span class="pre">m</span></code> is the length of the input along the axis specified by
<em class="xref py py-obj">axis</em>.</p>
</dd>
<dt><strong>axis</strong><span class="classifier">int, optional</span></dt><dd><p>Axis over which to compute the FFT. If not given, the last
axis is used.</p>
</dd>
<dt><strong>norm</strong><span class="classifier">{None, “ortho”}, optional</span></dt><dd><p>Normalization mode (see <a class="reference internal" href="../routines.fft.html#module-numpy.fft" title="numpy.fft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">numpy.fft</span></code></a>). Default is None.</p>
<div class="versionadded">
<p><span class="versionmodified added">New in version 1.10.0.</span></p>
</div>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>out</strong><span class="classifier">ndarray</span></dt><dd><p>The truncated or zero-padded input, transformed along the axis
indicated by <em class="xref py py-obj">axis</em>, or the last one if <em class="xref py py-obj">axis</em> is not specified.
The length of the transformed axis is <em class="xref py py-obj">n</em>, or, if <em class="xref py py-obj">n</em> is not given,
<code class="docutils literal notranslate"><span class="pre">2*m</span> <span class="pre">-</span> <span class="pre">2</span></code> where <code class="docutils literal notranslate"><span class="pre">m</span></code> is the length of the transformed axis of
the input. To get an odd number of output points, <em class="xref py py-obj">n</em> must be
specified, for instance as <code class="docutils literal notranslate"><span class="pre">2*m</span> <span class="pre">-</span> <span class="pre">1</span></code> in the typical case,</p>
</dd>
</dl>
</dd>
<dt class="field-odd">Raises</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>IndexError</strong></dt><dd><p>If <em class="xref py py-obj">axis</em> is larger than the last axis of <em class="xref py py-obj">a</em>.</p>
</dd>
</dl>
</dd>
</dl>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="numpy.fft.rfft.html#numpy.fft.rfft" title="numpy.fft.rfft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rfft</span></code></a></dt><dd><p>Compute the one-dimensional FFT for real input.</p>
</dd>
<dt><a class="reference internal" href="numpy.fft.ihfft.html#numpy.fft.ihfft" title="numpy.fft.ihfft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ihfft</span></code></a></dt><dd><p>The inverse of <a class="reference internal" href="#numpy.fft.hfft" title="numpy.fft.hfft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">hfft</span></code></a>.</p>
</dd>
</dl>
</div>
<p class="rubric">Notes</p>
<p><a class="reference internal" href="#numpy.fft.hfft" title="numpy.fft.hfft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">hfft</span></code></a>/<a class="reference internal" href="numpy.fft.ihfft.html#numpy.fft.ihfft" title="numpy.fft.ihfft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ihfft</span></code></a> are a pair analogous to <a class="reference internal" href="numpy.fft.rfft.html#numpy.fft.rfft" title="numpy.fft.rfft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rfft</span></code></a>/<a class="reference internal" href="numpy.fft.irfft.html#numpy.fft.irfft" title="numpy.fft.irfft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">irfft</span></code></a>, but for the
opposite case: here the signal has Hermitian symmetry in the time
domain and is real in the frequency domain. So here it’s <a class="reference internal" href="#numpy.fft.hfft" title="numpy.fft.hfft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">hfft</span></code></a> for
which you must supply the length of the result if it is to be odd.</p>
<ul class="simple">
<li><p>even: <code class="docutils literal notranslate"><span class="pre">ihfft(hfft(a,</span> <span class="pre">2*len(a)</span> <span class="pre">-</span> <span class="pre">2)</span> <span class="pre">==</span> <span class="pre">a</span></code>, within roundoff error,</p></li>
<li><p>odd: <code class="docutils literal notranslate"><span class="pre">ihfft(hfft(a,</span> <span class="pre">2*len(a)</span> <span class="pre">-</span> <span class="pre">1)</span> <span class="pre">==</span> <span class="pre">a</span></code>, within roundoff error.</p></li>
</ul>
<p>The correct interpretation of the hermitian input depends on the length of
the original data, as given by <em class="xref py py-obj">n</em>. This is because each input shape could
correspond to either an odd or even length signal. By default, <a class="reference internal" href="#numpy.fft.hfft" title="numpy.fft.hfft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">hfft</span></code></a>
assumes an even output length which puts the last entry at the Nyquist
frequency; aliasing with its symmetric counterpart. By Hermitian symmetry,
the value is thus treated as purely real. To avoid losing information, the
shape of the full signal <strong>must</strong> be given.</p>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">signal</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">fft</span><span class="p">(</span><span class="n">signal</span><span class="p">)</span>
<span class="go">array([15.+0.j,  -4.+0.j,   0.+0.j,  -1.-0.j,   0.+0.j,  -4.+0.j]) # may vary</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">hfft</span><span class="p">(</span><span class="n">signal</span><span class="p">[:</span><span class="mi">4</span><span class="p">])</span> <span class="c1"># Input first half of signal</span>
<span class="go">array([15.,  -4.,   0.,  -1.,   0.,  -4.])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">hfft</span><span class="p">(</span><span class="n">signal</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>  <span class="c1"># Input entire signal and truncate</span>
<span class="go">array([15.,  -4.,   0.,  -1.,   0.,  -4.])</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">signal</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mf">1.</span><span class="n">j</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mf">1.</span><span class="n">j</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">conj</span><span class="p">(</span><span class="n">signal</span><span class="o">.</span><span class="n">T</span><span class="p">)</span> <span class="o">-</span> <span class="n">signal</span>   <span class="c1"># check Hermitian symmetry</span>
<span class="go">array([[ 0.-0.j,  -0.+0.j], # may vary</span>
<span class="go">       [ 0.+0.j,  0.-0.j]])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">freq_spectrum</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">hfft</span><span class="p">(</span><span class="n">signal</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">freq_spectrum</span>
<span class="go">array([[ 1.,  1.],</span>
<span class="go">       [ 2., -2.]])</span>
</pre></div>
</div>
</dd></dl>

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